The Journal of Experimental Biology 207, 1715-1728 1715 Published by The Company of Biologists 2004 doi:10.1242/jeb.00947

Effects of distribution on the of level trotting in dogs David V. Lee*, Eric F. Stakebake, Rebecca M. Walter and David R. Carrier Department of Biology, University of Utah, 257 South 1400 East, Salt Lake City, UT 84112-0840, USA *Author for correspondence at present address: Concord Field Station, Department of Organismic and Evolutionary Biology, Harvard University, Bedford, MA 01730, USA (e-mail: [email protected])

Accepted 16 February 2004

Summary The antero–posterior mass distribution of quadrupeds loaded limb and (3) increase relative contact time of the varies substantially amongst species, yet the functional experimentally loaded limb, while (4) the individual limb implications of this design characteristic remain poorly mean fore–aft forces (normalized to body weight + added understood. During trotting, the forelimb exerts a net weight) would be unaffected. All four of these results were braking force while the hindlimb exerts a net propulsive observed when mass was added at the pelvic girdle, but force. Steady speed locomotion requires that braking and only 1, 3 and 4 were observed when mass was added at the propulsion of the stance limbs be equal in magnitude. We pectoral girdle. We propose that the observed relationship predicted that changes in body mass distribution would between antero–posterior mass distribution and individual alter individual limb braking–propulsive force patterns limb function may be broadly applicable to quadrupeds and we tested this hypothesis by adding 10% body mass with different body types. In addition to the predicted near the , at the pectoral girdle, or at results, our data show that the mechanical effects of the pelvic girdle of trotting dogs. Two force platforms in adding mass to the trunk are much more complex than series recorded fore- and hindlimb ground reaction would be predicted from mass distribution alone. Effects forces independently. Vertical and fore–aft impulses were of trunk moments due to loading were evident when calculated by integrating individual force–time curves and mass was added at the center of mass or at the pelvic Fourier analysis was used to quantify the braking– girdle. These results suggest a functional link between propulsive (b–p) bias of the fore–aft force curve. We appendicular and axial mechanics via action of the limbs predicted that experimental manipulation of antero– as levers. posterior mass distribution would (1) change the fore– hind distribution of vertical impulse when the limb girdles Key words: locomotion, running, dog, Canis, force, braking, are loaded, (2) decrease the b–p bias of the experimentally propulsion, center of mass, limb, trunk.

Introduction The function of individual limbs is difficult to assess in change as a constant proportion of the vertical component. running quadrupeds. Not only are the limbs in different Asymmetrical loading also occurs due to the natural variation positions with respect to the center of mass, they also interact in antero–posterior body mass distribution amongst simultaneously with the substrate. During steady speed quadrupeds. It is difficult to imagine similar forelimb forces in trotting, forelimbs typically exert a net braking force, while hyenas and baboons, for instance. Here we propose that body hindlimbs exert a net propulsive force. A common mass distribution alters both the vertical and fore–aft forces generalization that fore- and hindlimbs of quadrupeds exert exerted by the fore- and hindlimbs of trotting quadrupeds. similar forces (Alexander and Goldspink, 1977; Full et al., It is apparent that antero–posterior body mass distribution 1991) has fostered a lack of interest in individual limb varies substantially amongst quadrupeds, yet this characteristic mechanics of quadrupeds and has even led to the development has been measured during standing in fewer than 20 species of quadrupedal robots with fore- and hindlegs that function (Rollinson and Martin, 1981). In mammalian quadrupeds, the identically. How would such a robot carry an asymmetrically forelimbs typically support about 60% of body mass during distributed payload? Likewise, how would a tiger carry prey in standing, while the forelimbs of primates, lizards and alligators its jaws or a pregnant wildebeest carry its fetus? In these generally support less than 50%. Unfortunately, no data are examples, we expect that asymmetric loading will alter the available for quadrupeds with more extraordinary body types, fore–hind distribution of vertical force. This would be the only such as giraffes and spotted hyenas, which appear to have effect of asymmetrical loading if vertical and horizontal force extremely anterior center of mass positions, or rabbits, which components were independent. On the other hand, if the limbs appear to have extremely posterior center of mass positions. functioned as struts, the horizontal force component would We chose to study limb function during trotting because it 1716 D. V. Lee and others is the gait most commonly used by quadrupeds and it is a Table·1. Subject descriptions simple gait in which diagonal foot pairs are set down Mass Hip height Age alternately. This facilitates comparison of individual forelimb Dog (kg) (m) (years) Breed Sex and hindlimb function because diagonal fore- and hindfeet are A 29.5 0.545 8 Pointer/scent hound M set down in the same functional step. Trotting is unique in that B 34.5 0.573 2 German shepherd M the forelimb and hindlimb exert opposing fore–aft forces while C 23.5 0.525 5 Coonhound M in-phase with one another (Lee et al., 1999). The forelimb D 33.1 0.556 9 German shepherd mix M exerts a net braking force and the hindlimb exerts a net E 31.6 0.498 2 Golden retriever F propulsive force during each trotting step. Although individual limb data from trotting quadrupeds are limited to cats, macaques, dogs, goats, horses and alligators (Demes et al., 23.5 to 34.5·kg and from 2 to 9 years of age (Table·1). The 1994; Kimura, 2000; Lee et al., 1999; Merkens et al., 1993; dogs were borrowed from private owners for a period of 8·h Pandy et al., 1988; Rumph et al., 1994; Willey et al., 2004), or less and were never kept overnight. During data collection this phenomenon seems to be widespread in trotting animals. sessions, water was provided ad libitum and periodic rest Opposing fore–aft force has also been reported in hexapedaly breaks were given. The dogs trotted as they were led on a leash trotting cockroaches, which exhibit primarily foreleg braking across a level, hard-packed soil runway in which two force and hindleg propulsion (Full et al., 1991). This pattern reflects platforms were positioned in series (Fig.·1A). This allowed the substantial anterior inclination of the forelegs and posterior independent measurement of simultaneous fore- and hindlimb inclination of the hindlegs. In contrast, the mid-legs of ground reaction forces (Fig.·1B). cockroaches show no bias in limb angle and, accordingly, exert Data were collected while the dogs carried no load equal braking and propulsive force (Full et al., 1991). (unloaded, U) and under three loading conditions, in which In some cases, limbs act primarily as struts (Gray, 1968), tandem saddle bag packs were worn (Fig.·1A). Two small bags such that minimal moments are exerted about their proximal of lead shot totaling 10% body mass were inserted bilaterally joints. For example, the foreleg resultant force of trotting in the anterior compartments of the pectoral pack (fore-loaded, cockroaches tends to align closely with the leg axis (Full et F), the posterior compartments of the pectoral pack (mid- al., 1991). On the other hand, strut action is sometimes loaded, M), or the posterior compartments of the pelvic pack accompanied by substantial lever action (Gray, 1968). For (hind-loaded, H). We tested four hypotheses. Loading at the example, the strut action of the posteriorly inclined cockroach limb girdles (conditions F and H) will (1) change the fore–hind hindleg would exert higher propulsive force if not for the distribution of vertical impulse during trotting, (2) decrease the opposing lever action (i.e. proximal joint moment) tending to braking–propulsive (b–p) bias of the loaded limb and (3) protract the hindleg (Full et al., 1991). During steady speed increase the relative contact time of the loaded limb. (4) trotting, it would be advantageous for animals to use their Loading (conditions M, F and H) will not affect the weight- limbs as springy struts, minimizing proximal joint moments normalized mean fore–aft forces exerted individually by the and, thereby, increasing the overall economy of locomotion fore- and hindlimb. (Alexander, 1977). We predicted that body mass distribution would affect the Force and center of pressure measurements individual limb force patterns, and tested this idea by adding Force data were collected at 360·Hz from two strain gauge 10% body mass near the center of mass, at the pectoral girdle, type force platforms (made by N. T. Heglund) positioned or at the pelvic girdle of trotting dogs. Assuming that the limbs in series, using LabView software and a National act as struts, a disproportionate increase in loading of either the Instruments (Austin, TX, USA) data acquisition system fore- or hindlimb would increase both the vertical and fore–aft (DAQCard AI-16-E4, SCXI 1000 chassis, SCXI 1121 components of ground reaction force on that limb. Hence, in strain/bridge modules, and SCXI 1321 terminal blocks). the absence of functional compensation, hindlimb propulsive Data were collected for 2·s as the dogs crossed the force would tend to increase in response to pelvic girdle platforms. Only data from uninterrupted trotting were loading and forelimb braking force would tend to increase in saved. Each force platform was 0.6·m long by 0.4·m wide. response to pectoral girdle loading. Because braking and Using platforms of this length increased the likelihood that propulsion must be in equilibrium during steady speed trotting, diagonal fore- and hindfeet would strike separate force compensatory changes in limb function, as evidenced by the platforms simultaneously. Trials in which foot placements b–p bias and relative contact time, would be expected during did not meet this criterion were discarded. Furthermore, experimental loading of the limb girdles. footfalls that struck the platform edges as evidenced by negative vertical force (i.e. a moment tending to lift the opposite end) were discarded. The force platforms measured Materials and methods vertical and fore–aft ground reaction force (GRF) with Subjects and data collection separate double-cantilever transducers at each corner post. Five adult dogs (Canis familiaris L.) of various breeds were Vertical GRF acting upward and fore–aft GRF acting in the used in this experiment. Subjects ranged in body mass from direction of travel were considered positive. Vertical impulse Mechanics of level trotting in dogs 1717

and A Fy,hind =(jy,hind/tc,total)/mg·. (4) The mean angle of the resultant force vector during fore- or hindlimb contact was determined by: –1 θ = tan (jy/jz)·. (5) Forelimb vertical impulse was expressed as a fraction of total vertical impulse during paired diagonal supports. This quantity is referred to as the vertical impulse ratio:

R = jz,fore/jz,total·, (6) (Jayes and Alexander, 1978; Lee et al., 1999). In order to determine a limb’s functional bias toward braking or propulsion, a Fourier method (Hamming, 1973) adapted to the analysis of force–time curves by Alexander 1.5 and Jayes (1980) was used to quantify fore–aft force curve B shape. This method decomposes a complex waveform into Fore five simple sinusoids of progressively higher frequency (i.e. 1 shape complexity). These sinusoids are known as Fourier Hind terms and each term has a coefficient, the magnitude of 0.5 which indicates its influence on the shape of the waveform. Hind The five coefficients generated by this analysis are a1, b2, a3, Force (BW) Force 0 b4 and a5, where a indicates a cosine term and b, a sine term. Fore The lower frequency terms a1 and b2 define the basic –0.5 waveform, hence, by expressing a1 as a fraction of b2 (or 0.3 0.35 0.4 0.45 0.5 0.55 vice versa), the waveform shape can be described in Time (s) dimensionless terms (Alexander and Jayes, 1980). Here, Fig.·1. (A) Simultaneous fore- and hindlimb supports were recorded a1/b2 was used to quantify fore–aft force curve shape, which by two separate force platforms (thick lines) during trotting. is referred to as the braking–propulsive (b–p) bias. (B) Ground reaction forces were measured independently by the two Force–curve shapes defined by negative and positive values force platforms. Solid lines indicate vertical force and broken lines of a1/b2 are shown in Fig.·2. Negative values indicate a fore–aft force. BW, body weight. braking bias, positive values indicate a propulsive bias, and zero indicates a symmetrical force curve with no bias toward braking or propulsion. In this study, the b–p bias (a1/b2) was jz and fore–aft impulse jy were determined by numerical computed from individual limb fore–aft force curves. integration of GRF from an individual limb over the limb Center of pressure was measured in a conventional manner contact time (tc,fore or tc,hind) or both diagonal limbs over the by comparing the vertical force from independent transducer paired contact time tc,total of the diagonal limbs. A key to the elements at the fore and aft ends of a force platform. The two notation used throughout this report is provided in Appendix platforms were calibrated on a continuous metric scale to A. facilitate the computation of distance between supports on Normalized mean vertical force exerted on the center of separate platforms (Bertram et al., 1997). Center of pressure mass during paired diagonal supports was determined by: position for each foot was determined by force-averaging over the duration of foot contact. In other words, instantaneous F =(j /t )/mg·, (1) z z,total c,total centers of pressure were weighted according to the where m is body mass (or 110% of body mass under loaded conditions) and g is gravitational acceleration. Likewise, normalized mean fore–aft acceleration of the center of mass, a1/b2 = –0.5 a1/b2 = 0 a1/b2 = 0.5 + which is equivalent to force in dimensionless terms, was determined by: 0

Ay =(jy,total/tc,total)/mg·. (2) – Normalized mean fore–aft force exerted separately on the forelimb and hindlimb during paired diagonal supports were Fig.·2. Illustration of braking–propulsive biases defined by different determined by: values of the Fourier shape ratio a1/b2. a1/b2 = –0.5 indicates a braking bias, a1/b2 = 0 indicates no b–p bias, and a1/b2 = 0.5 Fy,fore =(jy,fore/tc,total)/mg·, (3) indicates a propulsive bias. 1718 D. V. Lee and others instantaneous vertical force values, summed, and then divided were synchronized by connection of a manual switch to a by the summation of vertical force over the time of contact. PEAK Event Synchronization Unit, which simultaneously This avoided the confounding effect of an extreme anterior foot marked a video frame and triggered force acquisition via a position during toe-off, for example, when the vertical force is breakout connector to the data acquisition card. The eye and quite small. During paired diagonal contacts, the fore- and the base of the tail were digitized in every other frame during hindfeet struck separate platforms allowing calculation of the paired diagonal contacts (i.e. tc,total) and their mean horizontal distance p between their mean center of pressure positions. The (y) positions were computed. Then, the mean y-positions of horizontal distance between fore- or hindlimb centers of the eye and the base of the tail were averaged to define a pressure and a kinematic mid-point (described in the following kinematic mid-point. This mid-point was used as a reference section) was also determined. for the horizontal positions of the fore- and hindlimb centers Mean forward velocity vy,step was determined directly from of pressure with respect to the trunk. Finally, the vertical the force record by a method similar to that of Jayes and position of the base of the tail with respect to the substrate Alexander (1978), except that time and distance parameters was used to approximate the mean hip height h during were computed from vertical force peaks rather than initial trotting, because it is near the greater trochanter (Table·1). foot contacts. The times tstep and distances d between Videographic measurements were taken from four subsequent forelimb supports and subsequent hindlimb approximately steady speed trials (i.e. –0.04 ≤ Ay ≤ 0.04) for supports were determined from the times of vertical force each dog under each loading condition. peaks and the corresponding center of pressure positions of each foot. Mean forward velocity vy,step is the ratio of d to Statistics tstep. Because the paired diagonal supports of interest were Comparisons between treatment conditions were made using preceded by a single forelimb support and followed by a either regression analysis or repeated measures ANOVA and single hindlimb support, forelimb and hindlimb values of Tukey–Kramer multiple comparison tests (InStat 2.0). The vy,step, dstep, and tstep were averaged to provide the best latter method was applied to mean vertical force, mean fore–aft estimates of these parameters. In order to account for size acceleration, vertical impulse ratio, mean forward velocity and differences between dogs, mean velocity was expressed as a Froude number, as well as the distance between diagonal Froude number: footfalls, p, and the distance between fore- or hind footfalls and – the kinematic midpoint between the eye and base of the tail. Froude =v /√gh·, (7) y,step For individual limb mean fore–aft force and b–p bias, which where g is gravitational acceleration and h is hip height as were functions of mean fore–aft acceleration, least-squares defined in the following section. The aforementioned step linear regression was used to compute the y-intercept (i.e. the period tstep is one half of the stride period during trotting. For steady speed condition) and its 95% confidence limits (Sokal consistency with existing literature, the stride period (2tstep) and Rohlf, 1995). The regressions of Fy,fore and Fy,hind on Ay was used to normalize individual foot and paired diagonal were linear, as was the regression of a1/b2,fore on Ay. The log contact times. The ratio of foot contact time to stride period is transformation of a1/b2,hind allowed the use of a linear termed the duty factor DF and was computed as: regression to determine its steady speed value. Values of other locomotor parameters were determined by DF = t /2t ·, (8) total c,total step simultaneously regressing the parameter of interest on mean fore–aft acceleration and mean forward velocity. If both DF = t /2t ·, (9) fore c,fore step multiple regression coefficients were significantly different and from zero (P<0.10), the multiple regression equation was used DF = t /2t ·. (10) hind c,hind step to predict the steady speed value of the locomotor parameter. In addition, the ratio of fore- to hindlimb contact time tc,f/tc,h If only one regression coefficient was significantly different was computed. The time of initial hindfoot contact with respect from zero (P<0.10), the non-significant independent variable to initial forefoot contact was also expressed as a fraction of was dropped from the model and least-squares linear the stride period. This quantity, referred to as the hindlimb regression was used to predict either the steady speed value of phase shift αhind, was computed as: the locomotion parameter or its value at the appropriate mean velocity. In cases where both multiple regression coefficients α =(t – t )/2t , (11) hind i,hind i,fore step were not significantly different from zero (P>0.10), the where ti,hind and ti,fore are initial contact times. A negative value sampled mean of the locomotor parameter was reported. of αhind indicates that hindlimb contact occurred before forelimb contact. Results Videographic measurements During paired diagonal supports, mean vertical ground A video camera with VCR (PEAK Performance reaction force Fz was greatest in the unloaded condition Technologies Inc., Centennial, CO, USA) acquired (Table·2A). The unloaded mean vertical force of 1.04 body 120·images·s–1 in lateral view. Force and video acquisition weights (BW) was significantly (P<0.05) higher than that of Mechanics of level trotting in dogs 1719

Table·2. Mean vertical force, mean fore–aft acceleration and vertical impulse ratio

Loading Fz Ay condition Dog (BW) (BW) RR0 (A) U A 1.037 0.0037 0.663 0.665 U B 0.987 0.0535 0.640 0.678 U C 1.075 0.0248 0.658 0.676 U D 1.055 –0.0155 0.626 0.615 U E 1.027 0.0142 0.596 0.606 U X 1.036±0.015 0.0161±0.0115 0.637±0.012 0.648±0.016

M A 0.998 –0.0330 0.693 0.670 M B 0.927 0.0097 0.671 0.678 M C 1.010 0.0200 0.684 0.698 M D 1.004 0.0093 0.637 0.644 M E 0.995 0.0042 0.617 0.619 M X 0.987±0.015 0.0020±0.0091 0.660±0.015 0.662±0.014

F A 0.976 –0.0323 0.712 0.689 F B 0.957 0.0050 0.682 0.685 F C 0.999 0.0211 0.700 0.715 F D 1.009 –0.0392 0.682 0.654 F E 0.976 0.0023 0.629 0.631 F X 0.983±0.009 –0.0086±0.0116 0.681±0.014 0.675±0.015

H A 1.013 0.0056 0.644 0.648 H B 0.974 –0.0034 0.628 0.626 H C 1.017 0.0158 0.629 0.640 H D 1.004 –0.0101 0.591 0.584 H E 1.005 0.0329 0.566 0.589 H X 1.002±0.008 0.0082±0.0076 0.612±0.015 0.617±0.013

(B) U vs. M –0.049*** –0.0141ns 0.024** 0.014ns U vs. F –0.053*** –0.0248ns 0.044*** 0.027** U vs. H –0.034** –0.0080ns –0.025*** –0.031** M vs. F –0.004ns –0.0107ns 0.021** 0.013ns M vs. H 0.016ns 0.0061ns –0.049*** –0.044*** F vs. H 0.019ns 0.0168ns –0.069*** –0.057***

Fz, mean vertical force; Ay, mean fore–aft acceleration; R, vertical impulse ratio; R0, R at steady speed. Sample means are given for each dog and loading condition (U, unloaded; M, mid-load; F, fore-load; H, hind-load). BW, body weight, or 110% body weight during experimental loading. See the text for parameter definitions. (A) values are means for each dog and group means ± S.E.M. for each loading condition. (B) Differences between means of each loading condition (e.g. ‘U vs. M’ was computed as M–U). Asterisks indicate the results of Tukey–Kramer tests comparing each loading condition to the others (***P<0.001, **P<0.01, *P<0.05, nsP>0.05). the three loaded conditions, which were between 0.98 and unloaded and loaded conditions were taken into account by 1.0·BW (Table·2B). Only the unloaded Fz was significantly predicting locomotor parameters at an intermediate vy,step of –1 different from one (P<0.05). Mean forward velocity vy,step 2.86·m·s whenever a parameter was significantly (P<0.10) was, on average, higher in the unloaded than in the loaded related to vy,step (Table·3). This analysis produced statistically conditions (U, 3.09·m·s–1; M, 2.81·m·s–1; F, 2.70·m·s–1; H, similar values of distance between subsequent footfalls d as –1 2.77·m·s ). These differences were significant (P<0.05) only well as step period tstep across all four loading conditions (U, in the F condition. When vy,step was normalized as a Froude M, F and H) (Table·4). Ground reaction forces reconstructed number (yielding treatment means between 1.17 and 1.35), the from locomotor parameters reported in this section and in statistical comparisons between loading conditions were the Appendix B illustrate steady speed GRF patterns at 2.86·m·s–1 same as before normalization. Velocity differences between for U, M, F and H conditions (Fig.·3). 1720 D. V. Lee and others

Table·3. Regression coefficients (±95% C.I.) for various 1.5 parameters versus mean fore–aft acceleration and mean U forward velocity Fore 1 vs. Ay vs. vy Hind Parameter (BW) (m·s–1) r2 0.5 d (m) Hind U ns 0.111±0.021 0.58 (BW) Force M ns 0.103±0.028 0.54 0 F ns 0.095±0.032 0.35 Fore H –0.495±0.424 0.114±0.048 0.32 –0.5 tstep (s) 1.5 U ns –0.033±0.007 0.54 M M ns –0.045±0.010 0.66 1 F ns –0.046±0.011 0.50 H –0.163±0.146 –0.040±0.017 0.52 0.5 αhind

Unsns (BW) Force M ns –0.046±0.032 0.16 0 F ns –0.042±0.023 0.17 H ns –0.042±0.034 0.11 –0.5 DFtotal 1.5 Unsns F M ns –0.034±0.024 0.16 F ns –0.031±0.013 0.25 1 H ns –0.020±0.019 0.08 0.5 DFfore U ns –0.027±0.022 0.07 Force (BW) Force M ns –0.049±0.025 0.26 0 F ns –0.047±0.016 0.35 H ns –0.052±0.023 0.29 –0.5 DF hind 1.5 U ns –0.037±0.011 0.34 M ns –0.019±0.019 0.08 H F –0.224±0.144 ns 0.13 1 H ns –0.015±0.018 0.05 tc,f/tc,h 0.5 Unsns

Mnsns (BW) Force 0 F 0.871±0.723 –0.103±0.071 0.13 H ns –0.083±0.074 0.09 θ –0.5 fore (deg.) 0 0.05 0.1 0.15 0.2 U 49.9±3.43 1.34±0.577 0.92 Time (s) M 47.3±8.21 ns 0.75 F 50.4±8.94 ns 0.66 Fig.·3. Fore- and hindlimb force curves reconstructed from steady H 40.0±10.1 ns 0.56 speed Fourier coefficients predicted at 2.86·m·s–1 (see Materials and methods and Appendix B) for each of the loading conditions (U, θ (deg.) hind unloaded; M, mid-load; F, fore-load; H, hind-load). Solid lines U 56.6±6.79 –1.23±1.14 0.77 indicate vertical force and broken lines fore–aft force. M 59.7±13.0 1.21±1.28 0.73 F 63.8±13.6 ns 0.57 H 71.7±14.3 ns 0.68 Mean fore–aft acceleration Ay was not significantly different Ay, mean fore–aft acceleration; vy, mean forward velocity. between U, M, F and H conditions (Table·2). Nevertheless, Only significant (P<0.10) regression coefficients are shown (ns, substantial inter-subject variation was observed, with Ay not significant.) individual means ranging from –0.039 to 0.054·BW. Thus, U, unloaded; M, mid-load; F, fore-load; H, hind-load. some dogs tended to speed up and others tended to slow down See the text for parameter definitions. as they crossed the force platforms. Because the vertical Mechanics of level trotting in dogs 1721

Table·4. Steady speed locomotion parameters during level trotting at 2.86·m·s–1 for each loading condition Loading condition Parameter U M F H d (m) 0.624±0.008 0.615±0.011 0.628±0.012 0.626±0.014 tstep (s) 0.219±0.003 0.217±0.004 0.222±0.004 0.220±0.005 αhind 0.001±0.009 0.012±0.012 0.016±0.009* –0.010±0.010* DFtotal 0.461±0.005 0.487±0.009* 0.485±0.005* 0.478±0.006* DFfore 0.451±0.009 0.476±0.009* 0.477±0.006* 0.455±0.007 DFhind 0.383±0.004 0.405±0.007* 0.389±0.005* 0.418±0.006* tc,f/tc,h 1.19±0.017 1.18±0.032 1.23±0.024* 1.09±0.022* θfore (deg.) –3.32±0.221 –3.43±0.307 –2.92±0.302* –3.30±0.362 θhind (deg.) 5.84±0.438 6.46±0.454* 5.70±0.459 5.14±0.512*

Values are ±95% C.I. Significant differences (P<0.05) from the unloaded condition are indicated by asterisks. U, unloaded; M, mid-load; F, fore-load; H, hind-load. See Appendix A and the text for parameter definitions.

Fig.·4. Steady speed values (±95% CI) of mean fore–aft U M F H force (filled bars) and braking–propulsive bias a1/b2 (open bars) for the loading conditions U, M, F and H (see text). Significant difference (P<0.05) from the unloaded condition U is indicated by an asterisk. Hind Fore Hind Fore Hind Fore Hind Fore * * (Table·2B). In agreement with Hypothesis 1, F and * H conditions significantly altered the fore–hind impulse distribution, while M maintained the natural impulse distribution.

–0.036 –0.31 –0.039 –0.40 –0.033 –0.32 –0.035 –0.41 In response to hind loading, a1/b2,hind decreased significantly (P<0.05; from 1.02 to 0.88) with 1.02 1.16 0.94 0.88

0.036 0.039 0.033 0.035 respect to U (Fig.·4). This reduction in propulsive * * bias is consistent with Hypothesis 2. Nonetheless, * fore-loading did not significantly reduce forelimb braking bias (Fig.·4). In response to mid-loading, = Mean fore–aft force Fy = Braking–propulsive bias a1/b2 a1/b2,fore decreased significantly (P<0.05) and a1/b2,hind increased significantly (P<0.05) with respect to U. These unexpected increases in forelimb impulse ratio R is a linear function of Ay, individual R values braking bias and hindlimb propulsive bias are considered in required adjustment to better predict the steady speed vertical the Discussion. impulse ratio R0. This was done according to the functional Across all four conditions, a1/b2,hind magnitude was, on relationship between R and Ay, reported to be –0.71 in separate average, 2.8 times that of a1/b2,fore (Fig.·4). In the unloaded analyses of Labrador retrievers and greyhounds, which span a condition, for example, a1/b2,fore was –0.31 and a1/b2,hind was full range of R0 from 0.56 to 0.64 (Lee et al., 1999). A discrete 1.02. Hence, the hindlimb showed a large propulsive bias and slope of the regression of R on Ay was not available from the the forelimb, a relatively small braking bias. Similar patterns present analysis due to disparity in body mass distribution were observed in conditions M and F. However, the fore–hind associated with the use of various breeds. Hence, the regression difference in b–p bias became much less pronounced in response coefficient of –0.71 was applied in the estimation of R0: to hind-loading, such that hindlimb propulsive bias was only twice the forelimb braking bias (Fig.·4). R0 = R + Ay(–0.71)·. (12) Like a1/b2,hind, the angle of the hindlimb resultant force θhind Mean values of R and R0 are reported for each subject in increased significantly (P<0.05) during mid-loading and Table·2. R0 was not significantly different between U and M, decreased significantly (P<0.05) during hind-loading with but fore-loading increased R0 by 0.027 (P<0.01) and hind- respect to unloaded trotting, while it was statistically loading decreased R0 by 0.031 (P<0.01) with respect to U unchanged during fore-loading (Table·4). In contrast to 1722 D. V. Lee and others θ a1/b2,fore, fore increased significantly (P<0.05) during fore- A loading but did not decrease significantly during mid-loading with respect unloaded trotting (Table·4). In agreement with Hypothesis 3, the relative contact time of the forelimb tc,f/tc,h increased significantly (P<0.05) during fore-loading and decreased significantly (P<0.05) during hind- loading (Table·4). During mid-loading, relative contact time was statistically unchanged (P>0.05) from the unloaded condition. Duty factors increased significantly (P<0.05) under all loading conditions, with the exception of DFfore during p hind-loading (Table·4). B Steady speed values of Fy,fore were negative (braking) and steady speed values of Fy,hind were positive (propulsive) under U all conditions (Fig.·4, filled bars). By definition, braking and propulsion were of equal magnitude during steady speed trotting. In the unloaded condition, for example, the magnitude M was 0.036 BW (i.e. 3.6% of body weight). In agreement with Hypothesis 4, loading conditions F and H were not F significantly different from U. Nonetheless, mid-loading resulted in an unexpected, statistically significant increase in H * braking and propulsive magnitude to 0.039·BW (P<0.05) (Fig.·4). 0 0.1 0.2 0.3 0.4 0.5 α During unloaded trotting, hindlimb phase shift hind was Distance (m) approximately zero, indicating simultaneous initial contacts of Fig.·5. (A) The kinematic mid-point between the eye and the base of the forelimb and hindlimb. Values of αhind were significantly greater (P<0.05) during fore-loading and significantly less the tail is represented by the vertical line posterior to the elbow. The distance p between hindlimb and forelimb centers of pressure was (P<0.05) during hind-loading (Table·4). In other words, measured from force platform data and was partitioned according to hindlimb contact was relatively later during fore-loading and this mid-point. (B) Distances from the mid-point to the hindlimb relatively earlier during hind-loading. This pattern is illustrated (open bars) and forelimb (filled bars) centers of pressure for the in the GRF reconstructions of Fig.·3. loading conditions U, M, F and H (see text). Error bars indicate 95% The distance between the supporting diagonal limbs p was confidence intervals. Significant difference (P<0.05) from the statistically unchanged (P>0.05) across loading conditions, unloaded condition U is indicated by an asterisk. with means of U, 0.569·m; M, 0.569·m; F, 0.562·m; H, 0.538·m. Hence, the spacing of diagonal supports was similar in U, M and F, while reduced, but not quite significantly, in H. good alignment of the added mass with the forefoot. The Position of the diagonal supports with respect to the kinematic predicted value is, however, smaller than the actual value in midpoint between the eye and base of the tail, was also the hind-loaded condition, suggesting that the hindfoot was conserved across U, M and F conditions; however, a somewhat posterior to the added mass, which rested near the statistically significant (P<0.05) anterior shift of the hindfoot iliac crests. Experimental loading near the center of mass position was observed during hind-loading (Fig.·5). The successfully maintained the natural distribution of vertical hindlimb reached about 0.03·m further forward in the hind- impulse between the fore- and hindlimb, as evidenced by the loaded than in the unloaded condition. statistically similar R0 of the unloaded and mid-loaded conditions. The dogs tended to trot faster without an added load, but Discussion the difference in mean forward velocity was statistically Experimental mass manipulations significant only between the unloaded and fore-loaded Experimental loading of the pectoral and pelvic girdles conditions. Multiple regression of locomotor parameters on increased the fraction of vertical impulse on the fore- and both mean fore–aft acceleration and mean forward velocity hindlimb, respectively, during trotting (Hypothesis 1). The (Tables 3 and 4) allowed the computation of steady speed –1 steady speed vertical impulse ratios R0 were 0.648 in the values at a common mean velocity of 2.86·m·s . This analysis unloaded condition, 0.675 during fore-loading, and 0.617 removed the potentially confounding effects of subtle velocity during hind-loading. An idealized model, in which the mass is differences between loading conditions. For example, the added directly above the supporting forefoot or hindfoot, distance between subsequent footfalls d was greater in the would predict R0 of 0.68 in the fore-loaded condition and 0.59 unloaded than in the loaded conditions when simply in the hind-loaded condition. The predicted value is quite close comparing mean values, but multiple regression analysis to the actual value in the fore-loaded condition, indicating a revealed that d was similar across the four loading conditions Mechanics of level trotting in dogs 1723 when compared at an intermediate speed of 2.86·m·s–1 Load (Table·4). Furthermore, step period tstep was similar to unloaded values across loading conditions (Table·4). This Hip Shoulder result agrees well with published stride periods of loaded and moment moment unloaded trotting in horses carrying 14% body mass at –1 4.0·m·s (Sloet van Oldruitenborgh-Ooste et al., 1995) and in Hypaxial –1 horses carrying 19% body mass at speeds above 3.0·m·s tension (Hoyt et al., 2000). In contrast to step period, duty factors were greater in the loaded conditions than in the unloaded condition. This agrees qualitatively with time of contact results from load-carrying experiments in trotting horses (Hoyt et al., 2000; Sloet van Oldruitenborgh-Ooste et al., 1995). Fig.·6. Trunk mechanics during trotting with a mid-trunk load. The load tends to ventroflex the trunk. Hindlimb retractors and forelimb Effect of adding mass near the center of mass protractors exert moments that tend to dorsiflex the trunk. Hypaxial On the basis that limbs act primarily as struts, we predicted muscle tension also tends to dorsiflex the trunk. that adding mass near the center of mass would not effect individual limb mean fore–aft force (normalized to body weight + added weight) (Hypothesis 4). Such a result was functional patterns when whole-body mechanics are previously observed when mass was added near the center of considered. Although exerting opposing moments and fore–aft mass of running humans, in which absolute braking and forces seems like a waste of metabolic energy, it may in fact propulsive impulse components increased in direct proportion be necessary to stabilize the trunk under certain circumstances. to the added weight (Chang et al., 2000). We also predicted Gray (1968) hypothesized that simultaneous forelimb that the individual limb b–p bias (i.e. the Fourier ratio a1/b2) protracting and hindlimb retracting moments would exert an would be affected by addition of mass at the limb girdles, but upward (dorsiflexing) bending moment on the trunk (Fig.·6). not when mass was added near the center of mass (Hypothesis He argued that this mechanism could reduce the tension 2). Nonetheless, both mean fore–aft force and b–p bias showed required in the hypaxial muscles that resist the downward increased magnitudes in the mid-loaded condition with respect (ventroflexing) bending moment due to . This effect can to the unloaded condition (Fig.·4). The magnitude of mean be demonstrated easily with a flexible 15·cm ruler. Imagine fore–aft force increased from 0.036 to 0.039 BW, while holding the ruler face up with thumb and forefinger of each forelimb b–p bias decreased from –0.31 to –0.40 and hindlimb hand at its ends. The ruler will sag, or bend downward, under b–p bias increased from 1.02 to 1.16. This unexpected increase its own weight, putting the bottom surface of the ruler in in both mean fore–aft force and b–p bias magnitudes has two tension. Now twist your palms upward – the resulting bending potential explanations. The first is a change in limb excursions moment causes the ruler to bend upward, putting the top such that the forefoot would be positioned more anteriorly and surface of the ruler in tension. The moments you applied to the the hindfoot, more posteriorly on average during stance. This ends of the ruler are analogous to the moments exerted on the would alter the mean fore–aft force and b–p bias due to the trunk by the muscles of the shoulder and hip. If mass were action of the limbs as struts with steeper antero–posterior added to the middle of the ruler, you would have to exert larger inclinations. Alternatively, the mean fore–aft force and b–p moments to bend the ruler upward. When mass is added at the bias could have been augmented by moments exerted by the center of mass of trotting dogs, our data show a compensatory protractor muscles of the shoulder and retractor muscles of the increase of ±0.53·N·m in shoulder and hip moments. hip. Such a mechanism would constitute a classic example of Therefore, the shoulder plus hip moment contributing to action of the limbs as levers, as described by Gray (1968). In upward bending of the trunk would be1.06·N·m. our experiments, a simple measurement of the horizontal By modeling the trunk as a beam rigidly fixed at its ends by distance p between simultaneous fore and hind supports the action of shoulder and hip muscles, one can determine the showed that this distance was the same in the unloaded and hip and shoulder moments required to maintain a net bending mid-loaded conditions (Fig.·5). Hence, fore- and hindfoot moment of zero across the length of the trunk. Estimating that positions could not have shifted in opposite directions. the distance between the shoulder and hip was p (0.569·m) and Forelimb protracting and hindlimb retracting moments must that the added load of 0.1·BW (29.9·N) was applied at a have been responsible for the observed changes in mean distance 0.66p anterior to the hip, the required shoulder plus fore–aft force and b–p bias during mid-loading. Assuming a hip moment would be 3.8·N·m. This suggests that the mean leg length of 0.54·m, both the shoulder and hip must have appendicular (i.e. shoulder and hip) muscles contributed just exerted additional mean moments of ±0.53·N·m to produce the 28% of the moment required to balance the downward bending additional ±0.003·BW of mean fore–aft force measured during moment due to the added load. It is likely, however, that the mid-loading. remaining upward bending moment was imparted to the trunk Opposing shoulder and hip moments reveal important by hypaxial muscles (Fig.·6). Because most hypaxials, such as 1724 D. V. Lee and others the rectus abdominis, act to bend the trunk upward, they could hindlimb initial contact increased from 0.001 (unloaded) to provide the additional bending moment required to support a 0.016 during fore-loading, indicating that hindlimb was set mid-trunk load. In a previous study, such hypaxial action was down 1.5% of the stride period later in the fore-loaded evidenced by increased electrical activity in the internal condition (Fig.·3, Table·4). The same trend has been observed oblique muscle of dogs trotting with mid-trunk loads (Fife et in Labrador retrievers and greyhounds, with hindlimb initial al., 2001). The use of appendicular muscles to exert a bending contact following forelimb initial contact in forelimb ‘heavy’ moment on the trunk is significant because it demonstrates a Labradors, but preceding forelimb initial contact in greyhounds functional link between axial and appendicular mechanics. (Bertram et al., 2000). Whether or not this pattern extends to Gray’s hypothesis aptly explains the observed effect of mid- other trotting quadrupeds is unknown. loading on fore- and hindlimb mean fore–aft force and b–p bias. Effect of adding mass at the pelvic girdle Just as addition of mass at the pectoral girdle increased Effect of adding mass at the pectoral girdle forelimb relative contact time (tc,f/tc,h), addition of mass at the We hypothesized that the loaded limb (the forelimb in this pelvic girdle decreased forelimb relative contact time (i.e., case) would show a decrease in b–p bias (Hypothesis 2). increased hindlimb relative contact time). In agreement with However, no statistically significant change in b–p bias was Hypothesis 3, tc,f/tc,h decreased from 1.19 (unloaded) to 1.09 observed in either of the limbs during fore-loading. The during hind loading (Table·4). As predicted (Hypothesis 4), forelimb braking bias was nearly identical in fore-loaded and mean fore–aft force was unchanged from that of unloaded unloaded conditions and the hindlimb propulsive bias showed trotting (Fig.·4). The most prominent and functionally a slight, insignificant decrease with respect to the unloaded important difference between unloaded and hind-loaded value (Fig.·4). The fore- and hindlimb kinematics were similar trotting was the b–p bias. We predicted that the propulsive bias in the fore-loaded and unloaded conditions, as evidenced by of the hindlimb would decrease in order to compensate for the distance between simultaneous fore and hind supports p increased vertical impulse due to hind-loading (Hypothesis 2). and the distances of fore and hind supports from the mid-point This would prevent hindlimb propulsion from dominating between the eye and base of the tail (Fig.·5). Hence, during forelimb braking. As expected, hindlimb propulsive bias steady speed trotting, the forelimb may act as a strut with little decreased significantly from the unloaded value of 1.02 to or no antero–posterior inclination. This would explain the 0.88 in the hind-loaded condition (Fig.·4). This compensatory absence of a reduced braking bias in response to increased change could have been achieved by decreasing the horizontal forelimb loading. As predicted (Hypothesis 4), mean fore–aft inclination of the hindlimb (strut action), decreasing the force was statistically unchanged from that of unloaded hindlimb retracting moment (lever action), or a combination of trotting. strut and lever action. From center of pressure data, it is clear In agreement with Hypothesis 3, the forelimb relative that strut action could account for the decrease in propulsive contact time tc,f/tc,h increased from 1.19 (unloaded) to 1.23 bias. During hind-loading, the hindlimb center of pressure was during fore-loading (Table·4). This follows the same pattern significantly (P<0.05) more anterior than during unloaded observed in Labrador retrievers and greyhounds, which trotting (Fig.·5). Given that this anterior shift was, on average, have substantially different antero–posterior body mass 0.03·m and the mean hindlimb length was 0.54·m, we know distributions. Although their hindlimb duty factors were that the mean hindlimb angle was approximately 3.2° retracted statistically similar, Labrador retrievers (R0=0.64) had with respect to the unloaded angle. forelimb duty factors of 0.505 and greyhounds (R0=0.56) had Without other changes in limb mechanics, a 3.2° change in forelimb duty factors of 0.426, a statistically significant mean limb angle would reduce propulsive force by 0.021·BW difference (P<0.05; Bertram et al., 2000). Hence, a relatively [i.e. 0.38·BW(tan3.2°)], which is seven times that required to high forelimb duty factor is naturally associated with a higher compensate for the effect of hind-loading (i.e. 0.003·BW). The fraction of vertical impulse on the forelimb. We propose that, observed repositioning of the hindfoot would have reduced the in general, relative fore- and hindlimb duty factors reflect the mean fore–aft force by more than half without a hindlimb antero–posterior mass distributions of trotting quadrupeds. retracting moment to maintain the observed mean fore–aft Biewener (1983) measured relative contact time of the force of 0.035·BW (Fig.·4). Given a fore–aft force deficit of forelimb in trotting mammals across a size range of 0.01 to 0.018·BW (i.e. 0.021–0.003) and a limb length of 0.54·m, the 270·kg. At the trot–gallop transition speed, these ratios were mean hindlimb retracting moment would need to be 2.9·N·m, less than 1.0 in mammals likely to support a larger fraction of about three times that exerted by the shoulder plus hip during body weight on their hindlimbs (i.e. pocket mouse, 0.88; mid-loading. During hind-loading, the pelvis itself seems to mouse, 0.99; chipmunk, 0.96; ground squirrel, 0.95), but have behaved much like a cantilever beam, supported greater than 1.0 in mammals known to support more body primarily at the hip (posterior pelvis) while mass was added weight on their forelimbs (i.e. dog, 1.07; pony, 1.04; horse, at the iliac crests (anterior pelvis). We conclude that the 1.01). observed changes in the partitioning of hindlimb strut and The fore-loaded condition also produced an unexpected lever action were largely due to local effects of loading the change in the timing of foot placement. The phase shift of pelvis. If the added mass were applied 0.10·m anterior to the Mechanics of level trotting in dogs 1725

Hindlimb Forelimb Fig.·7. (A) Fore–aft GRF of trotting dog with R0=0.65, based A upon the unloaded b–p biases (a1/b2) from Fig.·4 and duty factors (DF) from Table·4. (B) A hypothetical quadruped with a1/b2 > R0=0.50, showing equal fore- and hindlimb magnitudes of b–p R0=0.65 bias and equal duty factors. (C) A hypothetical quadruped with R0=0.35, showing a reversal of the fore- and hindlimb patterns DF 0.38 < 0.44 observed in trotting dogs.

B quadrupeds. For example, trotting dogs (R0=0.64) show a hindlimb propulsive bias approximately three times a1/b2 = R0=0.50 greater than the forelimb braking bias and a hindlimb duty factor 15% smaller than that of the forelimb. DF 0.41 = 0.41 Cheetahs (Acinonyx jubatus, R0≈0.52; Kruger, 1943), which have more symmetric fore–hind weight distributions, are predicted to show approximately equal C fore- and hindlimb b–p biases and duty factors. Lizards, which support more weight on their hindlimbs, are a1/b2 < predicted to show greater forelimb braking biases and R0=0.35 reduced forelimb duty factors relative to the hindlimb. This general pattern is illustrated in Fig.·7, which DF 0.44> 0.38 summarizes our results from trotting dogs and, based on these data, makes predictions for hypothetical quadrupeds with different fore–hind mass distributions. As shown in Fig.·7A,C, the limb that supports a greater hip joint, a hindlimb retractor moment of 2.9·N·m would be fraction of total vertical impulse is expected to have a greater required to balance its effect. This is consistent with the actual duty factor and a b–p bias closer to zero. The limb that placement of the added mass near the iliac crests. In contrast supports a smaller fraction of total vertical impulse is to mid-loading, the applied moment appears to have been expected to have a smaller duty factor and, due to its reduced resisted primarily by the hip muscles instead of being vertical force and contact time, is expected to compensate distributed between the shoulder, hip and, presumably, the with a greater b–p bias. These duty factor predictions are also trunk muscles. consistent with the tendency of more heavily loaded limbs to Finally, the forelimb braking bias increased from the be relatively longer. unloaded value of –0.31 to –0.41 in the hind-loaded condition Individual limb forces have been reported for only a handful (Fig.·4). The slight, statistically insignificant, anterior of trotting quadrupeds. Although most of these data represent movement of the forefoot and/or an increase in forelimb trotting with substantial net braking, the results generally protracting might have contributed to the observed support the pattern of limb function illustrated in Fig.·7. In increase in braking bias. Dutch warmblood horses, Merkens and coworkers (Merkens et al., 1993) reported a vertical impulse ratio of 0.55 and a b–p Implications for other quadrupeds bias similar to that of Fig.·7B. There was a small mean braking The axial and appendicular musculoskeletal systems force of –0.014·BW, on average, in their sample of trotting of quadrupeds are designed to accommodate specific steps. In cats, forelimb peak vertical force was found to be antero–posterior mass distributions. In this sense, the 1.69 times that of the hindlimb (Demes et al., 1994), which experimental addition of mass at discrete points seems suggests a vertical impulse ratio between 0.60 and 0.70 during unnatural. A load concentrated at the center of mass, pectoral trotting. The same study showed that forelimb braking girdle, or pelvic girdle is certainly not equivalent to the mass impulse was 3.5 times forelimb propulsive impulse and distributions inherent to different structural designs. An hindlimb propulsive impulse was 2.7 times hindlimb braking artificial load is, however, equivalent in one basic respect – impulse, indicating a larger b–p bias in the forelimb. This its effect on the center of mass position. Based upon pattern seems unusual in light of Fig.·7A, but is consistent the assumption that limbs act primarily as struts, we with the mean net braking acceleration of –0.037·BW in their hypothesized that a shift in antero–posterior mass distribution sample of trotting steps. The mean fore–aft forces exerted by would alter individual limb mechanics. For example, the the forelimb (–0.068·BW) and hindlimb (0.031·BW) of hindlimb propulsive bias decreased and its relative contact trotting cats are quite similar to those predicted for time increased in the hind-loaded condition. We propose that greyhounds trotting with a net braking acceleration of this relationship between antero–posterior mass distribution –0.037·BW (i.e. –0.070 and 0.033·BW, respectively; Lee et and limb function represents a general trend in trotting al., 1999). 1726 D. V. Lee and others

Even fewer trotting data are available from quadrupeds spotted hyenas or lions or, in other cases, chase a lone hyena with vertical impulse ratios below 0.50. Although individual from their kill at high speed (Creel and Creel, 2002). The limb function has been described in at least ten primate most ubiquitous natural increase in trunk loading occurs in species (Demes et al., 1994), most primates use asymmetrical gravid females, which often carry substantial loads over running gaits at higher speeds (Hildebrand, 1967). periods of weeks or months. For example, neonatal mass is Nonetheless, Kimura (2000) has collected fast trotting data approximately 15% of maternal mass in the pronghorn and from adult Japanese macaques Macaca fuscata. He reported springbok (Robbins and Robbins, 1979). This value neglects a vertical impulse ratio of 0.47, which is nearly equal to his the of the amniotic fluid and placenta, however, which measurement of standing weight distribution (0.46; Kimura are likely to be substantial. The distribution of fetal and et al., 1979). This is surprising, given that Kimura’s sample placental mass is generally posterior to the center of mass in of trotting steps represented a mean net braking acceleration quadrupeds. of more than –0.10·BW, on average (Kimura, 2000). In trotting dogs, this degree of braking would cause a substantial Conclusions increase in vertical impulse ratio (e.g. from 0.64 to 0.71 in Adding mass to the pectoral or pelvic girdles significantly Labradors; Lee et al., 1999). It is possible that macaques use altered the fore–hind vertical impulse distribution of trotting a mechanism, such as forward foot placement (Raibert, dogs. Assuming that the limbs act as struts, we predicted 1990), to avoid a nose-down pitching moment (and that these changes would lead to a decrease in b–p bias and subsequent vertical impulse redistribution) due to net an increase in relative contact time of the experimentally braking. Because forelimb braking impulse was 11.1 times loaded limb, while mean fore–aft force would be unaffected. forelimb propulsive impulse and hindlimb propulsive All three of these predicted results were observed in the impulse was roughly equal to hindlimb braking impulse, hind-loaded condition. Only the latter two were observed Kimura’s data suggest a b–p bias similar to that of Fig.·7C. in the fore-loaded condition, perhaps due to a smaller initial It is likely, however, that steady speed trotting of macaques b–p bias and/or limb inclination of the forelimb. We is more like Fig.·7B. propose that the observed relationships between antero- Some mammals with extreme antero–posterior mass posterior mass distribution, b–p bias, and relative contact distributions, such as giraffes, hyenas and rabbits, have time will apply to other quadrupeds. Our data also show that abandoned trotting and pacing gaits in favor of galloping and the mechanical effects of adding mass to the trunk are much bounding gaits (Estes, 1993; Howell, 1944; Pennycuick, 1975). more complex than would be predicted from center of mass This may also be the case for some primates (Demes et al., position alone. During mid-loading, the shoulder and hip 1994; Schmitt and Lemelin, 2002). Given that dogs (R0=0.64) moments increased in order to resist the downward bending have a fairly extreme hindlimb propulsive bias, a more anterior moment applied to the trunk. During hind-loading, the center of mass position, such as that of hyenas, may exceed a hindlimb retractor muscles exerted a large moment about the limit by which propulsive impulse can be supplied by hip to resist the moment applied to the pelvis. In accordance increasing hindlimb propulsive bias. Because hindlimb with the pioneering models proposed by Sir James Gray propulsive and forelimb braking impulses must be equal in (1968), both of these results exemplify a link between magnitude during steady speed trotting steps, extreme appendicular and axial mechanics via action of the limbs as antero–posterior mass distributions might favor galloping and levers. bounding over trotting and pacing. Although forelimb and hindlimb mechanics should generally accommodate a particular antero–posterior mass Appendix A distribution, an individual’s body mass distribution often Fz Dimensionless mean vertical force changes in response to behavioral or physiological factors. Fy Dimensionless mean fore–aft force Male cervids grow substantial masses at the end of a long Ay Dimensionless mean fore–aft acceleration neck every spring and then discard this mass in the θ Angle of the resultant force vector (degrees) autumn. The ‘Irish elk’ Megaloceros giganteus, with 40·kg R Vertical impulse ratio antlers estimated to be approximately 7% body mass, R0 Steady speed vertical impulse ratio provides a dramatic example (Geist, 1987). Carnivores p Distance between diagonal foot centers of pressure can carry heavy prey in their jaws for some distance (m) before consuming it. For example, man-eating tigers d Distance between fore- and hindlimb centers of have been reported to run while carrying an intact human pressure of adjacent steps (m) in their jaws (Corbett, 1946). Gray wolves Canis lupus tc Time of contact (s) often gorge on meals as large as 20–30% of their body tstep Time between fore- and hindlimb vertical force mass, yet they may subsequently travel a few miles to find a peaks of adjacent steps (s) –1 suitable spot to rest (Mech, 1970). After making a kill and vy,step Mean forward velocity (m·s ) gorging as much as possible, African hunting dogs Lycaon DF Duty factor pictus may need to escape from large scavengers such as αhind Hindlimb phase shift Mechanics of level trotting in dogs 1727

Appendix B Alexander, R. M. and Goldspink, G. (1977). Mechanics and Energetics of Animal Locomotion. London, New York: Chapman and Hall. Distributed by Steady speed Fourier coefficients during level trotting at Halsted Press. 2.86·m·s–1 Alexander, R. M. and Jayes, A. S. (1980). Fourier analysis of forces exerted Vertical force curves in walking and running. J. Biomech. 13, 383-390. Bertram, J. E. A., Lee, D. V., Case, H. N. and Todhunter, R. J. (2000). a1,fore b2,fore a3,fore Comparison of the trotting gaits of Labrador Retrievers and Greyhounds. (BW) (BW) (BW) Am. J. Vet. Res. 61, 832-838. Bertram, J. E. A., Lee, D. V., Todhunter, R. J., Foels, W. S., Williams, A. U 1.06±0.02 –0.071±0.014 0.051±0.008 and Lust, G. (1997). Multiple force platform analysis of the canine trot: a M 1.05±0.03 –0.025±0.015* 0.061±0.013 new approach to assessing basic characteristics of locomotion. Vet. Comp. F 1.08±0.02 –0.041±0.011* 0.064±0.012* Orthoped. Traumatol. 10, 160-169. H 1.02±0.03* –0.039±0.015* 0.062±0.011 Biewener, A. A. (1983). Allometry of quadrupedal locomotion: the scaling of X 1.05 –0.044 0.060 duty factor, bone curvature and limb orientation to body size. J. Exp. Biol. 105, 147-171. Chang, Y. H., Huang, H. W., Hamerski, C. M. and Kram, R. (2000). The independent effects of gravity and inertia on running mechanics. J. Exp. a1,hind b2,hind a3,hind Biol. 203, 229-238. (BW) (BW) (BW) Corbett, J. (1946). Man-eaters of Kumaon. New York: Oxford University U 0.708±0.016 –0.079±0.008 0.019±0.003 Press. M 0.640±0.010* –0.060±0.011* 0.010±0.006* Creel, S. and Creel, N. M. (2002). The African Wild Dog: Behavior, Ecology, and Conservation. Princeton, N.J.: Princeton University Press. F 0.636±0.015* –0.058±0.010* 0.013±0.004* Demes, B., Larson, S. G., Stern, J. T. J., Jungers, W. L., Biknevicius, A. H 0.706±0.021 –0.060±0.010* 0.011±0.006* R. and Schmitt, D. (1994). The kinetics of primate quadrupedalism: X 0.673 0.064 0.013 ‘hindlimb drive’ reconsidered. J. Hum. Evol. 26, 353-374. Estes, R. (1993). The Safari Companion: A Guide to Watching African Mammals: Including Hoofed Mammals, Carnivores, and Primates. Post Fore–aft force curves Mills, Vt.: Chelsea Green. Fife, M. M., Bailey, C. L., Lee, D. V. and Carrier, D. R. (2001). Function a1,fore b2,fore a3,fore of the oblique hypaxial muscles in trotting dogs. J. Exp. Biol. 204, 2371- U –0.052±0.004 0.149±0.005 –0.002±0.005 2381. M –0.055±0.006 0.143±0.006 –0.012±0.004* Full, R. J., Blickhan, R. and Ting, L. H. (1991). Leg design in hexapedal runners. J. Exp. Biol. 158, 369-390. F –0.045±0.005* 0.153±0.005 –0.007±0.004 Geist, V. (1987). On the evolution of optical signals in deer: a preliminary H –0.050±0.007 0.139±0.006* –0.004±0.005 analysis. In Biology and Management of the Cervidae (ed. C. M. Wemmer), X –0.051 0.146 –0.006 pp. 235-255. Washington: Smithsonian Institution Press. Gray, J. (1968). Animal Locomotion. New York: Norton. Hamming, R. W. (1973). Numerical Methods for Scientists and Engineers. a1,hind b2,hind a3,hind New York: McGraw-Hill. Hildebrand, M. (1967). Symmetrical gaits of primates. Am. J. Phys. U 0.087±0.005 0.075±0.004 0.004±0.003 Anthropol. 26, 119-130. M 0.085±0.005 0.069±0.005* 0.002±0.004 Howell, A. B. (1944). Speed In Animals; Their Specialization for Running and F 0.073±0.006* 0.070±0.003* 0.004±0.003 Leaping. Chicago: University of Chicago Press. H 0.076±0.007* 0.083±0.006* 0.005±0.005 Hoyt, D. F., Wickler, S. J. and Cogger, E. A. (2000). Time of contact and step length: the effect of limb length, running speed, load carrying and X 0.080 0.074 0.004 incline. J. Exp. Biol. 203, 221-227. Jayes, A. S. and Alexander, R. M. (1978). Mechanics of locomotion of dogs Values are ± 95% C.I. for each loading condition and overall (Canis familiaris) and sheep (Ovis aries). J. Zool. 185, 289-308. means are shown. Kimura, T. (2000). 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